Recent advancement of measurement techniques in the biomedical field has led to a considerable number of findings in the study of proteins, cells, and organs. Those findings are, however, specific to each particular class of objects with a particular relative size or scale. For this reason, the researchers have not necessarily been successful in some study fields such as interactive behavior of cells and organs, which fall into different scale ranges.
The computer technologies, on the other hand, have evolved rapidly toward ever higher performance, and computer simulation tools have taken advantage of such performance enhancements. Simulation-oriented approach has thus been increasingly used in expectation of new discoveries which could not be achieved by conventional measurement-based or experiment-based approaches, while compensating for insufficient accumulation of knowledge in the noted study fields.
The heart of humans is a popular subject of biomedical research using computer simulation techniques. The blood circulation is one of the fundamental mechanisms of life, in which the heart serves as an important organ that sends out blood to the entire body. It is noted that the heart itself also needs a stable supply of blood. Delivery of blood to the heart is done via a network of vessels known as the coronary circulatory system. The pathological conditions derived from abnormalities in the coronary circulation are collectively referred to as ischemic heart diseases. Ischemic heart diseases are mainly caused as a result of coronary arteriosclerosis.
The coronary circulation system has its own form of vessels, which begins with a large vessel running on the myocardial surface, branching into a plurality of distributing vessels with a slight reduction of their diameters. These vessels then run toward the heart chamber (in the transmural direction) while repeating bifurcation with a diameter reduction, thus forming delivering vessels. The coronary artery originates in the aortic sinus of Valsalva and repeats bifurcation and diameter reduction to become small arteries (several 100 μm in diameter) and arterioles (about 200 μm or smaller) until they reach the microcirculation system. At the opposite side of the microcirculation system, coronary venous vessels repeat joining and widening, thus forming venules and small veins extending toward the right atrium. The coronary arteries (and also the coronary veins for this matter) have a tree structure with a single largest vessel at the root and the microcirculatory vessels at the ends of branches. Some portions of coronary veins, however, have a bypass-like structure. The microcirculation system, including arterioles, venules, and capillaries, have a mesh (or network) structure to feed oxygen and nutrients to myocardial cells. The network structure of coronary vessels allows a variety of flows (e.g., parallel flows, counter flows) in the microcirculation system, so as to enable uniform blood supply to the myocardial cells.
The diameter of capillaries ranges from about 5 μm to 6 μm, which is smaller than the size of red blood cells. Red blood cells thus distort their shape during passage through the capillaries. It is the glycocalyx layer in endothelium that permits red blood cells to smoothly travel through vessels while touching their inner wall.
As mentioned, arteriosclerosis is the main cause of ischemic heart diseases, and most studies on arteriosclerosis have been made from the viewpoint of the biochemistry and cell biology. The pathogenesis of ischemic heart diseases is closely related to functional factors such as structure and control of the coronary circulatory system. The heart contraction is considered to give a strong effect on the coronary blood flow. It is known that, with the presence of angina pectoris, ischemia is likely to occur at the endocardial (heart chamber) side of the heart wall, where the tissue pressure increases most significantly. This suggests that the mechanical analysis, in combination with analysis of effects of blood flow (shear stress) related to atheroma formation, is expected to play an important role in study of the pathogenesis of ischemic heart diseases.
Such hemodynamics peculiar to the coronary circulation has drawn the interest of researchers including those who engaged in the basic medical sciences. They have made various kinds of experimental research and have accumulated many findings. One thing to consider here, however, is the fact that the coronary vessels have a wide range of diameters. For example, the diameters of trunks are in the order of millimeters, whereas those of capillaries are in the order of micrometers. That is, the largest coronary vessels are about 1,000 times as thick as the smallest coronary vessels. Another thing to consider is that the heart repeats contraction and relaxation to make a large motion of strokes, which adds significant constraints to realtime observation of coronary blood vessels. These difficulties have hampered deeper investigation of hemodynamics inside the heart wall.
To overcome the above difficulties, the efforts have been made to develop new techniques for experiment and diagnosis. In addition, computer simulation with a coronary circulation model enables observation of coronary vessels without being bound by the technical constraints noted above and is thus expected to be a powerful tool for illuminating the mechanisms of the heart.
The computer simulation uses a computational model as input data. A process of producing a model for simulation is called “modeling.” In the case of coronary circulation simulation, the modeling process reproduces, on a computer, a system of coronary vessels on the basis of anatomical and physiological findings about the coronary circulation. More specifically, a coronary circulation model is constructed by successively dividing each vessel segment with a particular diameter and length into multiple vessel segments, or joining multiple vessel segments into a single vessel segment, and then mapping the vessel segments to spatial coordinates of a heart. This modeling technique makes it possible to construct as small vessels as less than 1 mm in diameter, based on mathematical models. These vessels are so small that ordinary computed tomography (CT) scanners or magnetic resonance imaging (MRI) systems are unable to capture their images properly. One of the main purposes of the above studies is to elucidate the coronary hemodynamics and use the findings for prediction of diseases or curative effects. Development of a three-dimensional vascular network model for computer simulation takes part in those efforts.
The modeling methods being studied for coronary circulation simulation are largely divided into two types. One method first produces vascular data having no spatial coordinates, based on detailed anatomical data (e.g., statistics) describing the diameters, lengths, and branching structure of vessels, and then maps it to three-dimensional geometry of the heart. The other method, assuming that in-vivo vessels take a certain optimized form, models a vascular network by applying an optimal model within the geometry of the organ. Either method enjoys the advanced performance of computers to build a large-scale vascular model. The following description gives more specifics about these methods.
(a) Method Based on Anatomical Statistics Data
Input data for this method includes statistics on, for example, the coronary arteries of a swine heart which has been obtained through anatomy. Using the given statistics data together with random numbers generated, the method determines the lengths, diameters, and branching structure (interconnections) of vessels. The resulting vessels are then mapped on the finite element meshes of both ventricles.
(b) Method Based on Optimal Modeling
The method builds a coronary circulation model, without relying on anatomical data, but using an optimal model of vascular networks. The optimization means, in this case, minimizing the sum of energy for making the blood flow (assuming Hagen-Poiseuille flow) and energy for holding the blood (which is proportional to vessel volume). Since each organ has its own specific requirements for the amount of blood flow and the pressure difference for perfusion, the energy for blood flow takes a constant value. A model may then be constructed so as to minimize the volume of vessels under the noted constraints of the amount of blood flow and pressure difference. This model is called a minimum volume model. Based on the minimum volume model, there has been developed another method called the “constrained constructive optimization” (CCO). The CCO assigns flow rate distributions to the geometry of an organ, as well as specific blood pressures at the root and end points of the tree structure of vessels. The CCO then produces new vessel segments at the end points, connects them to other vessel segments, and further moves their connection points so as to minimize the total volume of vessels according to the law of the Hagen-Poiseuille flow.
See, for Example, the Following Documents:
G. S. Kassab, C. A. Rider, N. J. Tang and Y. C. Fung, “Morphometry of Pig Coronary Arterial Trees,” American Journal of Physiology, Heart and Circulatory Physiology, 265: H350-H365, 1993
G. S. Kassab, D. H. Lin and Y. C. Fung, “Morphometry of pig coronary venous system,” American Journal of Physiology, Heart and Circulatory Physiology, 267: H2100-H2113, 1994
G. S. Kassab and Y. C. Fung, “Topology and dimensions of pig coronary capillary network,” American Journal of Physiology, Heart and Circulatory Physiology, 267: H319-H325, 1994
N. P. Smith, A. J. Pullan and P. J. Hunter, “Generation of an anatomically based geometric coronary model,” Annals of Biomedical Engineering, vol. 28, pp. 14-25, 2000
B. Kaimovitz, Y. Lanir and G. S. Kassab, “Large-scale 3-D geometric reconstruction of the porcine coronary arterial vasculature based on detailed anatomical data,” Annals of Biomedical Engineering, vol. 33(11), pp. 1517-1535, 2005
B. Kaimovitz, Y. Lanir and G. S. Kassab, “A full 3-D reconstruction of the entire porcine coronary vasculature,” American Journal of Physiology, Heart and Circulatory Physiology, vol. 299, pp. H1064-H1076, 2010
C. D. Murray, “The physiological principle of minimum work,” I. Proceedings of National Academy of Science, 12 207, 1926
A. Kamiya and T. Togawa, “Optimal branching structure of vascular tree,” Biophysics, vol. 13 No. 2, pp. 77-81, 1973
R. Karch, F. Neumann, M. Neumann and W. Schreiner, “A three-dimensional model for arterial tree representation, generated by constrained constructive optimization,” Computers in Biology and Medicine, vol. 29, pp. 19-38, 1999
W. Schreiner, R. Karch, M. Neumann, F. Neumann, P. Szawlowski and S. Roedler, “Optimized arterial trees supplying hollow organs,” Medical Engineering and Physics, vol. 28, pp. 416-429, 2006
Z. L. Jiang, G. S. Kassab and Y. C. Fung, “Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree,” Journal of Applied Physiology, vol. 76, pp. 882-892, 1994
However, the aforementioned coronary circulation model produced on the basis of anatomical statistics data tends to contain some portions that does not match well with anatomical characteristics of the organ that it represents. For example, a cardiac model produced in this way may have some regions exhibiting an unnaturally low density of vessels, whereas it is known that there has to be a noticeable number of vessels.
The method based on optimal modeling has evolved to be able to produce a partial model of an organ (e.g., a single ventricle with a simplified geometry), but it is not mature enough to reproduce anatomical points as to the entire coronary circulation. For example, the method is still incapable of modeling the left coronary circumflex artery (LCX) and right coronary artery (RCA) running along the boundaries between ventricles and atriums.
The above section has discussed difficulties in producing a coronary circulation model. The researches will, however, encounter similar challenges when modeling a vascular network of other organs.